Suppose that the starting salary offered to graduates who majored in psychology is $47,507.

Assume that X, the starting salary offer for psychology majors, is normally distributed with a mean of $47,507 and a standard deviation of $5,000.
A friend tells you that his friend, a psychology major, received a starting salary offer of $52,000. You think this number sounds high and wonder whether your friend was mistaken. Use the logic of hypothesis testing to evaluate whether you believe your friend was mistaken.

The z score of a starting salary of $52,000 is (?)****   .

The probability that a psychology major received a starting salary offer higher than $52,000 is**** (?)

Using a significance level of .05, you  (can or cannot)****   conclude that your friend is mistaken

Let's break this problem down step by step using the hypothesis testing logic.

### **Step 1: Hypotheses**
We are testing whether the salary of $52,000 is unusually high based on the information provided.

- **Null Hypothesis (H₀):** The salary offer of $52,000 is consistent with the normal distribution of psychology majors' starting salaries (mean = $47,507, standard deviation = $5,000).
- **Alternative Hypothesis (H₁):** The salary offer of $52,000 is unusually high and inconsistent with the normal distribution of psychology majors' starting salaries.

### **Step 2: Calculate the Z-score**
The **Z-score** measures how many standard deviations a given value is from the mean. The formula for the Z-score is:

\[
Z = \frac{X - \mu}{\sigma}
\]

where:
- $X = 52,000$ (the salary offer),
- $\mu = 47,507$ (the mean salary),
- $\sigma = 5,000$ (the standard deviation).

Let's calculate the Z-score.

\[
Z = \frac{52,000 - 47,507}{5,000}
\]The Z-score for a starting salary of $52,000 is approximately **0.90**.

### **Step 3: Calculate the Probability**
Next, we find the probability that a psychology major receives a starting salary higher than $52,000. We can do this by calculating the cumulative probability up to a Z-score of 0.90 using a standard normal distribution table or function, and then subtracting it from 1.

Let's find this probability.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, I can explain how to find the probability manually using a Z-table or calculator. For a Z-score of 0.90, you would look up the cumulative probability from a standard normal distribution table, which gives the probability up to a Z of 0.90 (approximately 0.8159). To find the probability that a psychology major's salary is higher than $52,000:

\[
P(X > 52,000) = 1 - P(Z < 0.90) = 1 - 0.8159 = 0.1841
\]

Thus, the probability is approximately **0.1841** (or 18.41%).

### **Step 4: Conclusion Using Significance Level**
The significance level ($\alpha$) is given as 0.05 (or 5%). Since the probability of receiving a salary higher than $52,000 is **18.41%**, which is greater than 5%, you **cannot** conclude that your friend is mistaken. The offer of $52,000 is not significantly unusual.

Would you like more details or have any questions?

Here are five related questions to explore further:
1. How does changing the standard deviation affect the Z-score?
2. What if the salary offer were $55,000? Would the conclusion change?
3. What is the Z-score for a salary offer of $45,000?
4. Can you explain Type I and Type II errors in hypothesis testing?
5. How would you interpret a significance level of 0.01?

**Tip:** In hypothesis testing, the significance level ($\alpha$) defines how strict you want to be when rejecting the null hypothesis.

Suppose that the starting salary offered to graduates who majored in psychology is $47,507. Assume that X, the starting salary offer for psychology majors, is normally distributed with a mean of $47,507 and a standard deviation of $5,000. A friend tells you that his friend, a psychology major, received a starting salary offer of $52,000. You think this number sounds high and wonder whether your friend was mistaken. Use the logic of hypothesis testing to evaluate whether you believe your friend was mistaken. The z score of a starting salary of $52,000 is (?). The probability that a psychology major received a starting salary offer higher than $52,000 is (?). Using a significance level of .05, you (can or cannot) conclude that your friend is mistaken.

Evaluate whether a psychology major's starting salary of $52,000 is unusually high using hypothesis testing. Learn how to calculate the Z-score and probability for normally distributed data with a mean of $47,507 and a standard deviation of $5,000. Suitable for students in Grades 10-12.

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